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ERIODIC  CONJUGATE  NETS 


A  DISSERTATION 

presented  to  the 

Faculty  of  Princeton  University 

IN  Candidacy  for  the  Degree 

OF  Doctor  of  Philosophy 


BY 


EDWARD  S.  HAMMOND 


Reprinted  from  the  Annals  of  Mathematics,  Second  Series,  Vol.  22,  No.  4,  June,  1921 


Accepted  by  the  Department  of  Mathematics, 
May,  1920 


PERIODIC  CONJUGATE  NETS 


A  DISSERTATION 

presented  to  the 

Faculty  of  Princeton  University 

IN  Candidacy  for  the  Degree 

OF  Doctor  of  Philosophy 


BY 

EDWARD  S.  HAMMOND 


Reprinted  from  the  Annals  of  Mathematics,  Second  Series,  Vol.  22,  No.  4,  Juno,  1921 


a  3- 


[Reprinted  from  Annals  of  Mathematics,  Vol.  XXII.,  No.  4,  June,  1921.] 


PERIODIC   CONJUGATE    NETS.'    ""*•    '^^*o.... 

By  Edward  S.  Hammond. 

Introduction.  If  n  functions  of  u  and  v,  x''^\  x''"\  •  •  •,  z'-"\  which 
satisfy  an  equation  of  Laplace  of  the  form 

.  .  dH    _  d  log  a  dx       d  log  h  dx 

dudv  dv      du  du      dv  ' 

be  interpreted  as  the  homogeneous  coordinates  of  a  surface  in  (n  —  1) 
space,  the  parametric  curves  on  this  surface  are  said  to  form  a  conjugate  net. 
Where  no  ambiguity  arises,  this  system  of  curves  or  the  surface  on  which 
it  lies  will  be  called  simply  the  net  A^".  Equation  (1)  will  be  called  the 
point  equation  of  A''.     Now  the  functions*  Xi  and  X-i,  given  by 

.  .  _  dx       d  log  a  _  dx       d  log  b 

are  also  homogeneous  coordinates  of  nets,  Ni  and  A''-!,  which  are  called 
the  first  and  minus  first  Laplace  transforms  of  N.  Ni  has  as  its  first  and 
minus  first  Laplace  transforms  nets  A^2  and  N  itself;  A^2  is  called  the 
second  Laplace  transform  of  N.  Developing  these  transforms  in  both 
senses  we  get  a  series  of  nets  •  •  •  N's,  •  •  •,  N-i,  N,  Ni,  •  -,  Nr,  •  •  •, 
called  a  sequence  of  Laplace. f  This  sequence  will  be  called  the  sequence 
Nr.  In  the  first  section  of  this  paper  general  properties  of  this  sequence 
will  be  developed. 

In  section  2,  we  impose  upon  the  sequence  Nr  the  condition  that  it  shall 
be  periodic;  that  is,  that  a  certain  Laplace  transform  Np  of  A'^  shall 
coincide  with  A^  itself.  After  transformation  of  parameters  it  is  shown 
that  the  identity  of  A^p  and  A^  involves  the  identity  of  A^p_i  and  A''-! 
and  in  general,  of  N p-k  and  A^-^,  k  =  0,  1,  2,  •  •  •,p.  Necessary  conditions 
on  the  coefficients  of  the  point  equation  of  A^  are  derived  and  it  is  shown 
by  discussion  of  the  completely  integrable  systems  of  partial  differential 
equations  involved  that  these  conditions  are  also  sufficient.  It  is  also 
shown  that  if  an  equation  of  Laplace  of  form  (1)  is  the  point  equation  of 
one  periodic  net,  it  is  the  point  equation  of  an  infinity  of  others  of  the 
same  period. 

The  remainder  of  the  paper  is  taken  up  with  other  sequences  of  Laplace 

*  Here  xi  indicates  any  or  all  of  a;i<i\  Xi^'^'>,  . . . ,  a;/"\     A  similar  usage  is  followed  throughout, 
t  Darboux,  Logons  sur  la  thcoric  generale  des  surfaces,  2d  ed.  (1915),  vol.  II,  chap.  2. 

238 


4  4  4  o  n  r> 


239  EDWARD    S.    HAMMOND. 

closely  related  to  the  sequence  Nr.  The  sequences  studied  in  section  3 
involve  certain  properties  of  families  of  lines  in  higher  ordered  spaces 
which  we  pi:ocee<;i  to  develop.  The  lines  joining  corresponding  points 
df  a.'n^iyV  aad  its  first  Laplace  transform  A^i  form  a  two-parameter  family 
G,  each  line  of  which  is  a  common  tangent  of  these  surfaces.  Consider 
for  the  sake  of  definiteness  the  line  joining  the  points  on  A^  and  A^'i  with 
parameters  Uo  and  Vq.  Through  this  line  pass  two  developable  surfaces 
all  of  whose  generators  are  lines  of  G,  namel}'',  the  tangent  surfaces  of  the 
curve  u  =  Uoon  N,  and  of  the  curve  v  =  Vo  on  A^i.  WTien  a  two-parameter 
family  of  lines  in  higher  ordered  space  possesses  either  of  these  equivalent 
properties,  namely,  that  each  line  of  the  family  is  a  common  tangent  to 
two  surfaces,  and  that  through  each  line  there  pass  two  developable 
surfaces  all  of  whose  rectihnear  generators  are  lines  of  the  family,  it  is 
called  a  congruence.  In  3-space  any  two  parameter  family  of  lines 
possesses  these  properties,  but  in  space  of  higher  order  this  is  not  the  case. 
The  surfaces  to  which  all  the  lines  of  G  are  tangent  are  called  the  focal 
surfaces  and  the  nets  A^  and  A^i  the  focal  nets  of  the  congruence. 
Levy*  has  shown  that  the  functions  ^  and  v,  defined  by 

dv  du 

where  6  is  any  solution  of  (1),  may  be  interpreted  as  the  coordinates  of 
nets  which  will  be  called  Levy  transforms  of  A^  by  means  of  d.  The  points 
of  these  nets  lie  on  the  lines  joining  the  corresponding  points  of  A^  and 
A^i,  A'_i  and  A^,  respectively,  and  the  developables  of  the  congruences  so 
generated  cut  the  surfaces  of  the  nets  in  the  curves  of  the  nets.  In  section 
3,  it  is  shown  that  these  nets,  there  called  A^o,  i  and  A^_i.  o,  are  Laplace 
transforms  of  one  another.  It  is  also  shown  that  A^o,  i  is  a  Levy  transform 
of  A^i  by  means  of  6i,  a  solution  of  the  point  equation  of  A^i  formed  from  6 
by  the  same  process  by  which  the  coordinates  of  A^i  were  formed  from 
those  of  A'^.  From  these  properties  follows  a  very  intimate  connection 
between  the  two  sequences  of  Laplace,  Nr,  the  original  sequence,  and 
Nr,r+i  of  which  A^_i,  0  and  A^o,  i  are  two  nets.  The  sequence  A^,r+i  is 
called  the  first  Levy  sequence.  On  it  may  be  formed  a  first  Levj^  sequence, 
A^,  r+2  which  is  called  a  second  Levy  sequence  of  A^.  The  treatment  given 
in  section  4  of  these  sequences  and  of  the  Lev}-  sequences  of  higher  orders 
which  are  analogously  formed,  indicates  their  close  dependence  upon  the 
Laplace  transforms  of  A^.  They  are  actually  the  sequences  of  derived 
nets  of  higher  orders  studied  by  Tzitzeicaf  and  others. 

*  Levy,  Journal  dc  I'Ecolc  Polytechnique,  Vol.  LVI  (1S8G),  p.  67. 
t  Tzitzeica,  Coinptes  Rendus,  vol.  156  (1913),  p.  375. 


PEEIODIC    CONJUGATE    NETS.  240 

In  section  5,  the  results  of  section  2  are  applied  to  these  Levy  sequences 
and  conditions  for  their  periodicity  are  derived.  Two  interesting  geom- 
etric configurations  arising  under  special  conditions  are  discussed. 

As  a  property  of  the  Levy  transforms  of  a  net  A^,  it  was  mentioned 
that  the  developables  of  the  congruences  of  tangents  to  the  parametric 
curves  of  A^  cut  the  surfaces  of  the  Levy  transforms  in  the  curves  of  the 
nets.  Whenever  this  relation  holds  between  a  congruence  and  a  net, 
they  are  said  to  be  conjugate.  Two  nets  conjugate  to  the  same  congruence 
are  said  to  be  in  relation  T  and  the  transformation  which  carries  one  such 
net  into  the  other  is  called  a  transformation  T,  to  use  the  terminology  of 
Eisenhart*  who  has  developed  a  general  theory  of  such  transformations. 
The  congruence  to  which  both  nets  are  conjugate  is  called  the  conjugate 
congruence  of  the  transformation.  In  section  6,  it  is  shown  that  similar 
Laplace  transforms  of  two  nets  in  relation  T  are  also  in  relation  T,  and 
hence  that  two  sequences  of  Laplace  may  be  developed  such  that  corre- 
sponding nets  of  these  sequences  are  in  relation  T.  The  problem  of 
finding  a  sequence  Nr  so  related  to  the  original  sequence  Nr  is  reduced  to 
the  problem  of  finding  a  solution  0  of  the  adjoint  equation  of  (1)  and 
quadratures.  Owing  to  arbitrary  constants  arising  in  the  quadratures, 
their  integration  gives  a  multiple  infinity  of  such  sequences  between 
which  certain  geometric  relations  exist. 

The  results  of  section  2  are  then  applied  to  these  sequences,  and  it  is 
found,  first,  that  if  equation  (1)  has  periodic  solutions,  so  has  its  adjoint; 
second,  if  such  a  solution  0  be  used  in  the  determination  of  Nr,  the  se- 
quences Nr  are  also  periodic  of  period  p. 

1.  Sequences  of  Laplace.  In  the  study  of  these  sequences,  two  func- 
tions of  the  coefficients  of  equation  {!),  H  and  K,  defined  by 

_        5-  log  ad  log  a  d  log  b 
dudv  dv         du 

(4) 

_        3-  log  b      d  log  a  d  log  b 

dudv  dv  du  ' 

are  of  constant  occurrence.  Their  most  important  property  is  in  connec- 
tion with  the  transformation  to  other  coordinates  x',  such  that 

(5)  X  =  Xx', 

where  X  is  a  function  of  u  and  v.  Since  the  coordinates  x  are  homogeneous, 
evidently  this  transformation  has  no  effect  on  the  net.  The  coordinates 
x'  do  not  satisfy  equation  (1),  however,  but  are  solutions  of 

*  Eisenhart,  Trans.  Amer.  Math.  Soc,  vol.  18  (1917),  p.  97. 


241  EDWARD    S.    HAMMOND. 

d^d         d  ,      add   ,     d  ,      b  dd 

=  —  los; log  — ■ 

dudv       dv     '^X  du       du     ^\  dv 

(6) 

\       dudv     ^  dv     '^Xdu     ^\  dv  du  /    ' 

as  may  be  shown  by  differentiation.  If  the  functions  H  and  K  be  formed 
from  the  coefficients  of  (6)  and  the  resulting  expressions  reduced,  it  is 
found  that  they  are  identical  With  (4),  that  is,  H  and  K  are  invariant 
under  the  transformation  (5).  They  are  called  the  Laplace-Darboux 
invariants  of  the  equation  (1)  or  of  the  net  A'^.  If  the  independent  variables 
are  changed  by  a  transformation 

(7)  U  =  (/.M,  V  =  yp{v'), 

the  invariants  H'  and  K'  of  the  new  equation  are  given  by 

(8)  H'  =  <i>'{u'W{v')H,        K'  =  4>'{u'W{v')K, 

where  0'  and  ip'  are  the  first  derivatives  of  ^  and  4/  with  respect  to  their 
arguments. 

Consider  the  coordinates 

dx      d  log  a 

(9)  ^'  =  a".  -  ~^ir  ^' 

of  the  net  A^i  mentioned  in  the  introduction.     If  we  differentiate  with 

respect  to  u,  we  get 

/im  ^^1      a  log  6 

(10)  -d^  - -du- ""'  =  ^""^ 

a  relation  confirming  the  statement  of  the  introduction  that  the  lines 
joining  corresponding  points  of  A''  and  A'"!  are  tangent  to  the  curves 
v  =  const,  on  A^i.  Then  if  H  vanishes  equations  (9)  and  (10)  reduce  the 
solution  of  equation  (1)  to  quadratures;  also  in  this  case,  the  surface  A^ 
degenerates  into  a  curve.  But  if  H  does  not  vanish,  we  differentiate  with 
respect  to  v  and  find  that  the  coordinates  Xi  satisfy  the  equation  of  Laplace 

d^d   _  d  log  aH  d_e      d_\ogh  dd 

dudv  ~         dv        du  du      dv 

,f_d^,      h      d  log  aH  a  log  6      a  log  a  3  log  6  \ 

'^{dudv^^^a  dv  du     +      dv         du     +^;^' 

which  proves  that  A^i  is  also  a  net,  as  stated  in  the  introduction.  This 
equation  has  invariants  Hi  and  Ki,  analogous  to  H  and  K,  defined  by 

d^    ,      a-H  ,   a  log  a  a  log  6   ,  „         d'~    ,      aH 

dudv     ^    b  dv  du  dudv     °   b 

(12) 

Ki  =  —  ^-^  log  a  H 5 — -  ~z \-  c  =  H. 

dudv     ^  dv  du 


PERIODIC    CONJUGATE    NETS.  242 

Since  frequent  use  is  to  be  made  of  the  point  equation  of  nets  associated 
with  a  net  hr.ving  the  point  equation  (1),  for  the  sake  of  brevity  we  denote 
such  an  equation  by  the  expression 

(13)  [Xi]   Gi,  hi,  d], 

which  means  that  the  coordinates  Xi  of  the  net  Ni  satisfy  the  equation 

JiL  -  ^^^±i  ^  4_  ^  log  ^  ^ 
dudv  dv       du  du       dv 

(14) 

/  ^Mog  Cj  _  d  log  a^  d  log  b ,       ajog^  d  log  h         \ 

V     dudv  dv  du  dv  du  }    ' 


Also  the  net  N i  has  invariants 
(15) 


dudv     ^  Ci  dv  du  ' 


d'~    .      bid  log  a  a  log  6   , 
log  -  +  —^ ^t;-  +  c. 


'  dudv     ^  Ci  dv  du 

In  this  notation  (11)  becomes 

(16)  [xi]  aH,  b,  b/a], 

and  the  effect  of  the  transformation  (5)  on  the  point  equation  is  expressed 
by 

^  ^  iJ'  X'  \'  X  J' 

The  minus  first  Laplace  transform,  N-i,  is  the  second  focal  surface 
of  the  congruence  of  tangents  to  the  curves  v  =  const,  of  A^.  For,  by 
the  definition  of  its  coordinates  given  in  equation  (2),  the  lines  joining 
corresponding  points  are  tangent  to  N,  and  the  equation  obtained  by 
differentiating  these  coordinates  with  respect  to  v  and  using  (1)  shows 
them  to  be  tangent  to  A_i.     The  point  equation  of  this  net  is  denoted  by 

(18)  [a:_i;   a,  bK,  a/b]. 

Consider  now  the  congruences  of  tangents  to  the  parametric  curves  of 
Ni.  We  have  seen  that  the  congruence  of  tangents  to  the  curves 
V  =  const,  has  N  and  Ni  as  its  focal  nets;  that  is,  N  is  the  minus  first 
Laplace  transform  of  iVi.  This  is  also  obvious  as  a  consequence  of  equa- 
tion (10),  whose  left  member  is  the  expression  for  the  coordinates  of  the 
minus  first  Laplace  transform  of  Ni  formed  by  analogy  with  (2). 

The  second  focal  Surface  of  the  congruence  of  tangents  to  the  curves 
u  =  const,  of  Ni  is  the  net  No,  the  second  Laplace  transform  of  N.     Its 


243  EDWARD    S.    HAMMOND. 

coordinates 

,     ,  dxi      d  log  aH 

or,  using  (9), 

_  d-x      d  log  a^H  dx       (  alogg  d  log  aH  _  d-  log  a  \ 
^^^~d?~        dv        dv^\     dv  dv  dv-      J^' 

satisfy  the  equation  denoted  by  [.r2;  ciHHi,  h,  6-/a-i7]. 

Continuation  of  this  process  in  both  the  positive  and  negative  senses 
gives  the  nets  of  the  sequence  Nr. 

The  coordinates  of  the  rth  Laplace  transform  A^  are 


(20)              -  =  1r- 

a  log  aHHi  • 

dv 

•   •    Mr—'> 

Xr-1, 

or,  by  repeated  substitution, 

(21)  Xr    =   J^^r  +    ^^r,  r-l  ^  ^^,  +     --■    +Ar,  O-T. 


h^ 


where  the  Ap,  g  are  functions  of  a,  H,  Hi,  ■  ■■,  Hr-i  and  their  derivatives. 
The  point  equation  of  Nr  is 

(22)  yxr;  aHHi  ■  ■  ■  Hr-„  b,  ^Tjj^fj ^^-^  •  •  •  Hr-2  J  ' 

The  equations  analogous  to  (10)  and  (19)  are 

/OON  ^^'^  TJ  ,      a  log  6  dXr   _    a  log  Or 

^^^^  du  ^  ^'-'"^--^  +  -^du^'^-'  dv  ~       dv      "-'-  +  '''+" 

and  they  will  be  used  as  formulas  for  the  partial  derivatives  dXridu  and 
dXr/dv. 

On  the  negative  side  of  the  sequence,  the  general  Laplace  transform 
N-s  has  coordinates 

_  a.r_,-+i      a  log  hKK-i  •  ■  •  K-s+2 
^-'  ^  ~du  du  ^-'+'' 

or 

d^r  a*~H'  dr 

(24)  X.,  =  ^„.  +  S., ._,  ^^  +  . . .  +  B.. ,  ^  +  B,. .., 

which  satisfy  the  equation 

x-s',  a,  hKK-i  ■  •  •  K.s+i,  ^»j^s-i  . . .  j^_^^^  J  • 
The  formulas  corresponding  to  (23)  are 

dx-s  .    a  log  h-s  dx-s       a  log  a 


PERIODIC    CONJUGATE    NETS.  244 

From  (23)  and  (26)  it  follows  that  if  Hr-i  or  K_s+i  vanishes,  the  sequence 
terminates;  for  the  surface  Nr  or  N-s  degenerates  into  a  curve.  This 
is  a  special  case  of  great  importance*  but  it  is  not  before  us  in  this  paper. 
2.  Periodic  Sequences  of  Laplace.  In  the  introduction,  a  periodic  se- 
quence was  defined  as  a  sequence  such  that  a  certain  net  Np  coincided 
with  the  original  net  A^.  When  this  is  the  case,  the  coordinates  Xp  and 
X  must  satisf}^  the  relation 
(27)  Xp  =  \(u,  v)x, 

where  X  is  a  function  of  u  and  v  at  most,  and  is  the  same  for  all  ?i  coordi- 
nates.    The  coordinates  Xp  satisfy  the  equation  denoted  by 


(28) 


W' ''™'  •  ■  •  "■"'' '''  «?ffi=r^^TH;:;k] 


this  result  being  obtained  when  r  in  (22)  is  replaced  by  p.     From  (17), 
(27),  and  (28),  the  coordinates  x  must  satisfy 


(29) 


r      aHHi-^-JIp_,    h  h^  1 

L^'  X  'X'  aPR^-'  •••  Hp.i\y 


as  well  as  the  fundamental  equation  (1).  Since  in  every  case  which  we 
shall  consider  there  are  at  least  three  coordinates  x,  the  coefficients  of 
dx/du,  dx/dv,  and  x  in  (29)  and  in  (1)  must  be  equal.     We  have  therefore 

,oAN             ^1      aH  ■  ■  ■  Hp-i      a  log  a  d  b       dlogb 

(30)  ^log ^^ ^    -^     ,         -log-=--^^, 

From  the  equations  (30),  we  get 

alogX  ^  d  log  HHi  •  •  •  gp_i  alogX 

^  ^^  dv  dv  '  du  ' 

and  from  these 

Using  (32)  and  (33)  in  (31),  we  get 

(34)  a^„  log  a'H'-^'-  ■  ff ,_,  =  ^' 

which  can  also  be  obtained  immediately  from  the  equality  of  H,  the  in- 
variant of  (1),  and  Hp,  the  corresponding  invariant  of  (28). 

*  Darboux,  I.e.,  p.  33. 


245  EDWARD    S.    HAMMOND. 

Equation  (33)  may  be  integrated,  giving 

HH,  ■  ■  ■  H,.,  =  UV, 

where  U  and  V  are  functions  of  u  and  v  alone  respectively. 

From  equation  (8)  we  recall  the  effect  of  the  transformation  (7)  on  the 
invariants  H  and  K.     Likewise  under  this  same  transformation 

H/  =  <t>'{u'W{v')Hi. 

By  giving  to  i  values  from  0  to  79  —  1  and  multiplying,  we  get 

H'H,'  ■  •  ■  ^,_/  ==  HH,---  H,.A4>'4^'Y  =  U{u')VW)[<i>'^p'V, 

where  U  and  V  are  the  transforms  of  U  and  T'  under  (7).  Hence  0  and  \p 
may  be  determined  so  that 

H'Hi   •  •  •  H p-i    =1; 

then  from  (32),  X  equals  a  constant,*  w,  since 

d  log  X  _  ^ogj^  _ 
du  dv 

In  the  remainder  of  this  section,  we  shall  assume  that  this  transformation 
has  been  made,  dropping  primes  for  convenience. 

After  this  change  of  variable,  there  are  two  necessary  conditions  for  a 
sequence  of  Laplace  of  period  p,  namely 

(36)  HHiH.  ■  •  •  //p-i  =  1 

and  equation  (34).  To  show  that  these  conditions  are  sufficient,  we  pro- 
ceed as  follows.     Differentiate 

(37)  Xp  =  mx 

with  respect  to  u.     Using  (23),  (2),  and  (37),  we  find 

(38)  Hp-iXp-i  =  771X-1, 

which  states  analytically  the  fact,  evident  from  geometrj^,  that  if  Np 
coincides  with  N,  then  A^p_i  coincides  with  A^_i.  Differentiating  the 
last  equation  with  respect  to  u,  and  using  (23)  and  (26),  we  have 

d  log  hHp-i  ,        d  log  bK 

(39)  Hp-iHp-2Xp-2  +  Hp-i       '  -Q-—       ^P-i  =  ^"'^-2  +  m  -  ^^ —  X-i. 

Now  K  =  H-i  and  i/_i  =  i/p-i,  since  (38)  is  a  transformation  of  the 
type  (5).  The  equality  of  K  and  H p-i  may  also  be  derived  from  (34)  and 
(36),  using  the  values  of  these  invariants  given  by  (15).     Then  by  (38), 

*  Tzitzeica,  Comptes  Rendus,  vo.l  157  (1913),  p.  908. 


PERIODIC    CONJUGATE    NETS.  246 

equation  (39)  reduces  to 

(40)  Hp-iHp-2Xp-2  =  mx_2. 

If  we  continue  this  process  we  have  in  general, 

(41)  Hp-iHp-2  •  ■  ■  Hp-iXp-i  =  mx-i, 
or,  by  (36), 

(42)  Xp-i  =  7nHHi  ■  •  •  Hp^i-iX^i 
and  finally 

(43)  X  =  nix-p 

showing  that  A^  is  identical  with  its  minus  pth.  Laplace  transform  as  well 
as  with  the  pth  transform.  We  observe  that  this  process  is  reversible, 
that  is,  by  starting  from  (43),  differentiating  with  respect  to  v,  and 
reducing  step  by  step,  we  may  reproduce  this  same  set  of  equations. 

If  we  refer  to  (21)  and  (24)  it  is  evident  that  the  p  -\-  1  equations  given 
by  (41)  when  i  takes  integral  values  from  0  to  p  inclusive,  are  a  system  of 
linear  partial  differential  equations  of  various  orders  which,  with  (1), 
must  be  satisfied  by  the  coordinates  of  the  fundamental  net  A^  of  a  periodic 
sequence.  From  this  point  of  view,  let  us  examine  in  detail  the  trans- 
formation from  equation  (38)  into  (40),  as  this  is  entirely  typical  of  the 
change  from  any  one  of  (41)  into  the  next.  The  substitutions  for 
dXp-i/du  and  dx^i/du  from  (23)  and  (2)  first  engage  our  attention.  The 
value  of  dXp-i/du  used  depends  on  the  definition  of  Xp-^  and  on  the  use 
of  the  point  equation  of  Np-2.  But  this  point  equation  is  essentially  the 
result  of  differentiating  (1)  p  —  2  times  with  respect  to  v,  a  fact  which 
becomes  evident  on  consideration  of  the  result  of  substituting  the  value 
of  Xp-2  from  (21)  in  the  point  equation  denoted  by  (22).  The  value  of 
dx-i/du  used  is  merely  the  definition  of  the  minus  first  Laplace  trans- 
form. The  rest  of  the  reduction  may  be  based  as  indicated  on  the  two 
equations  (34)  and  (36).  From  these  considerations  and  from  the  re- 
versibility of  the  process  we  conclude  that,  by  virtue  of  (1),  its  derivatives, 
and  the  conditions  (34)  and  (36),  any  one  of  equations  (41)  or  (42)  is 
equivalent  to  any  of  the  others. 

If  the  period  be  odd,  let  us  set  p  =  2n  -\-  1,  and  i  =  n  in  (42),  so  that 
it  becomes 

Xn+l    =    mHHi    •  •  •    HnX-n- 

Also  by  setting  p  =  2n  +  1  and  i  =  n  +  1  in  (41),  we  get 

X—n—l    =    ~"  ll2ntl2n—\    '  '  '    tlnXn- 

The  differential  equations  to  which  these  are  equivalent  give  values  of 


247  EDWARD    S.    HAMMOND. 

d"+^:c/(9y"+^  and  <9"+^x/5i6''+^  in  terms  of  the  2n  +  1  or  p  quantities 


d''x     d"x     d"-^x     d"-'^x  dx     dx 

dv/" '   ay"'   dvT-^ '   d  v"-'^ '    '  '  ' '   du'   dv 


^,  X. 


All  other  derivatives  of  order  n  +  1  may  be  obtained  in  terms  of  these 
same  p  quantities  by  differentiation  of  (1).  Similarly,  when  the  period  is 
even,  let  p  =  2n  and  i  =  n,  n  +  1,  giving  the  two  equations 

Xn    =    mHHi    ■  ■  •    Hn-lX-n, 
X—n—1    —    ~  tl  2n-\tl  -211-2    '   '   '    H  n-lX n—\. 

By  means  of  these  equations  and  (1)  all  derivatives  of  the  «th  order  but 
d^'x/du'',  and  all  derivatives  of  higher  orders  may  be  expressed  in  terms 
of  the  2n  or  p  quantities 

d"x     a"~^x     d'^-'^x  dx     dx 

ai?"   du"-' '   ai'"-^'    '"'   du'  Jv'  ^' 

In  either  case  we  have  a  completely  integrable  system  of  equations  which 
possesses  but  p  independent  solutions.  From  this  result  follows  the 
theorem  stated  by  Tzitzeica: 

A  sequence  of  Laplace  of  period  p  can  exist  in  space  of  no  higher  order 
than  p  —  1. 
In  particular  we  note: 

The  only  nets  of  period  three  are  planar  nets. 

It  will  be  observed  that  the  conditions  (34)  and  (36)  do  not  involve  the 
constant  m.  Neither  is  it  involved  in  the  above  discussion  of  the  com- 
plete integrability  of  equations  (1)  and  (37).  Again,  the  equations  them- 
selves show  us  that  m  is  a  significant  constant,  that  is,  one  which  cannot 
be  reduced  to  unitj^  by  any  change  of  parameter.  Then  the  solutions  of 
the  system,  which  are  the  coordinates  x  of  our  fundamental  net  N,  may 
be  written  x'  {u,  v;  m),  i  =  1,2,  •  •  • ,  p. 

If  we  replace  m  by  another  constant,  m' ,  so  that  we  have  the  equation 
Xp  =  m'x',  instead  of  (37),  this  equation  forms  with  (1)  another  com- 
pletely integrable  system  with  p  independent  solutions,  which  we  may 
call  x'  {u,  v;  m').  A  similar  set  may  be  obtained  for  every  value  of  the 
constant.     We  may  state  this  result  as  follows: 

//  an  equation  of  Laplace  he  the  point  equation  of  a  net  whose  Laplace 
sequence  is  periodic  of  period  p,  it  is  the  point  equation  of  an  infinity  of  nets 
having  the  same  propcrti/. 

3.  Levy  sequences.  The  first  Levy  sequence.  In  equation  (3)  of  the 
introduction,  functions  ^  and  rj  are  defined  as  the  coordinates  of  the  Levy 


PERIODIC    CONJUGATE    NETS. 


248 


transforms  of  N  by  means  of  a  solution  d  of  the  point  equation  (1).  For 
the  study  of  these  transforms  in  connection  with  the  Laplace  sequence, 
it  is  advantageous  to  denote  the  coordinates  by  Xq,  i  and  a;_i,  o,  defined  by 


^0,1 


1 


X 

Xi 


X-1,  0    — 


1 


as  they  indicate  by  their  form  that  the  points  given  by  any  parameter 
values  lie  on  the  line  joining  the  points  of  N  and  its  Laplace  transforms 
with  the  same  parameters.     The  functions  6i  and  d^i,  defined  by 


(44) 


dd 
dv 


d  log  a 

dv 


dd_ 
dli 


d  log  b 
du 


are  called  the  first  and  minus  first  Laplace  transforms  of  6  and  are  solutions 
of  the  point  equation  of  Ni  and  A^_i  respectively.     As  we  have  the  relations 


P'^O,  1    — 


ay"' 


7_i.r_i,  0 


du 


V, 


the  functions  Xo,  i  and  :r_i,  o  differ  from  ^  and  rj  only  by  factors  of  pro- 
portionality, and  consequently  are  coordinates  of  the  Levy  transforms. 

We  shall  accordingly  denote  the  Levy  transforms  by  A^o,  i  and  A^_i,  ol 
their  point  equations  are  denoted  by 


(45) 
and 


^0,  i] 


I 


adi 


ad 


b, 


Let  the  net  Nr,  r+i  be  defined  by  its  coordinates  Xr,  r+i  namely. 


(46) 


-T,  r+1 


r  Xr 

r+1       ^r+l 


where  r  is  any  positive  or  negative  integer  or  zero,  and  where  di  is  formed 
from  d  by  (21)  and  (24)  as  Xi  is  formed  from  x.  The  order  of  the  sub- 
scripts in  Nr,  r+1  indicates  that  the  points  of  these  nets  are  to  be  considered 
as  situated  on  the  tangents  to  the  curves  u  =  constant  of  the  net  Nr', 
that  is,  on  the  line  between  any  net  Nr  and  its  positive  Laplace  transform, 
Nr+i.  The  application  of  (23)  and  (26)  to  nets  of  this  type  leads  to  the 
theorem : 

Any  Laplace  transfor7n  Nr  of  a  net  N  has  the  nets  Nr-i,  r  and  Nr,  r+i  as 
its  Levy  transforms  by  means  of  Or,  the  rth  Laplace  transforrn  of  a  solution  6 
of  the  point  equation  of  N;  or,  Nr,  r+i  is  a  Levy  transform  of  Nr  by  means  of 
dr,  and  of  Nr+i  by  means  of  6r+i. 


249  EDWARD    S.    HAMMOND. 

Let  us  express  the  coordinates  of  the  first  Laplace  transform  of  A^^-i,  o, 
following  (2)  and  simplify  them.     We  find 

dx-i  a       d  .       ad  _ 

~dv        'd'v  ^""^  ~eZ,  ""-'• '  -  ^'"-  '' 

that  is,  the  Levy  transforms  of  a  net  by  means  of  the  same  solution  of  its 
point  equation  are  Laplace  transforms  of  one  another. 

From  the  last  two  theorems  A^_i,  r  and  Nr,  r+i  are  Laplace  transforms 
of  one  another  for  every  value  of  r.  Then  A>,  r+u  (r  =  •  •  •,  —  2,  —  1,  0, 
1,2,  •  •  • ),  is  a  sequence  of  Laplace;  it  will  be  called  the  first  Levy  sequence. 
In  the  expressions  for  the  point  equations  and  the  formulas  for  the  partial 
derivatives  of  the  coordinates  of  the  nets  of  this  sequence,  it  is  necessary 
to  distinguish  between  positive  and  negative  subscripts.  If  r  and  s 
be  positive  integers,  we  have 

r  aHH,  ■  ■  •  Hr-idr+i         i)^+'  1 

r       — ^  hk^j^     K    ^ 1 

dU         -   tlr-X,rXr-Ur+  ^^        Xr.r+l, 

dXr,  r+1  d   log  ar,  r+1 


dv  dv 


^r,  r+1    ~r   •'^V+1,  r+-2', 


dx-s-i.-s      K^s-i.-s  ,    (9  log  6_s_i. 


du        ~      K-s      ''-'-'-'-'   '  du 

dx-s-i,  -s  _  d  log  g-s-i.  _ 
dv  dv 


,-.-1+  ^^  X-s-l,~S, 

X—s—l,  —s    ~\~    ■ts.  —  s+lX—s,  — s+1- 


4.  The  second  Levy  sequence.  Levy  sequences  of  higher  orders.  The  first 
Levy  sequence  is  built  up  from  the  fundamental  sequence  of  Laplace  by 
the  use  of  a  solution  6  of  (1)  and  its  Laplace  transforms.  On  this  Levy 
sequence  which  is  itself  a  sequence  of  Laplace,  we  may  build  a  second  Levy 
sequence  and  so  on  indefinite!}'. 

Let  ^0. 1  be  a  solution  of  equation  (45).  The  Levy  transforms  of  A^o,  i 
by  means  of  this  solution  are  the  nets  A^o.  2  and  A^_i,  1,  whose  coordinates 
are  defined  in  accordance  with  (46)  as  follows, 


1 

^0, 2 


'0, 1     ^0, 1 
'1, 2     •'^1, 2 


X-i.  1  = 


/-1, 0     X-i,  0 
h,  1        ^0,  1 


'0,  1 
where  61,2  is  the  first  Laplace  transform  of  ^o,  1,  and  ^_i,  0  is  its  minus 


PERIODIC    CONJUGATE    NETS. 


250 


first  transform  divided  by  /f_i,  o,  a  quantity  which  occurs  similarly  in 
X-i,  0-     The  point  equations  of  A^o,  2  and  A^_i,  1  are  denoted  by 

,.-,       r  adid,,2     .       &'    1  r  joedo.i      ,         ab      1 

^^^^     L'^°' ''  T^TT '  ^'  ^lo7i  J '       L^'-^-  ^'  0:7:0 '  ^'  OITo  J  • 

The  same  pair  of  theorems  which  established  the  first  Levy  sequence  and 
the  fact  that  it  is  a  sequence  of  Laplace  are  valid  here.  We  denote  by 
Nr,  r+2  and  N-s-2,  -s  the  general  nets  of  the  second  Levy  sequence  and 
give  their  coordinates,  namely 


1 


■r,  r+2 


-s—2,  -s    — 


'r,   r+1 
1 


'r,  r+1      ^r,   r+l 
'r+1,  r+2    ^r+1,  r+2 


-s—2,  — s— 1 


-s-2,  — s— 1  I  t'-s— 1, 


X—s—2,  — s-1 
•^—s—l,  — s 


Using  the  second  Levy  sequence  and  a  solution  of  (47),  a  third  Levy 
sequence  may  be  formed.  We  shall  not  give  the  details  of  this  sequence 
but  pass  at  once  to  the  A;th  or  general  sequence.  Here  the  net  correspond- 
ing to  A^o,  1  and  A^o,  2  is  A^o,  U  Its  coordinates  and  point  equation  and  the 
accompanying  differentiation  formulas  can  be  written  down  by  analogy 
with  the  corresponding  expressions  for  A'o,  1  and  A^o,  2,  and  their  accuracy 
established  by  induction.  Similar  methods  may  be  applied  in  the  study 
of  the  other  nets  of  the  general  sequence.  The  coordinates  of  the  general 
net  A^r,  r+k  are  defined  by 


•^r,  r+  k 


'r,  r+k— I 


r,  r-{- k — 1        3^r,  r+A;— 1 
r+1,  r+k       ^r+1,  r+k 


where  r  is  any  positive  or  negative  integer  or  zero,  and  k  any  positive 
integer. 

In  forming  the  second  Levy  sequence,  we  made  use  of  a  solution  ^o.  i 
of  equation  (45);  we  now  investigate  the  nature  of  this  function.  Sup- 
pose 6'  to  be  a  solution  of  (1)  such  that  there  is  no  linear  relation  con- 
necting 6,  6'  and  the  coordinates  x.     If 


(52) 


1 


'0,  1   — 


then  ^0, 1  is  a  solution  of  (45).  It  will  now  be  proved  that,  conversely,  to 
a  solution  ^o,  i  of  (45),  not  linearly  dependent  on  the  coordinates  Xo,  i, 
there  corresponds  a  solution  6'  of  (1)  linearly  independent  of  the  x's  and 
of  6.  Consider  the  net  ^o,  i  as  the  projection  in  {n  —  1)  space  of  a  net 
A^o.  1  in  ?z-space  whose  coordinates  are  3:0.1^^',  ^0,  i^~\  •  •  •,  ^0,  i^'*\  ^0,  i- 
Then  the  congruence  G,  composed  of  the  lines  joining  corresponding  points 


251 


EDWARD    S.    HAMMOND. 


of  N  and  Ni,  is  the  projection  of  a  congruence  G  in  n-space  conjugate  to 
the  net  A^o,  i-  One  of  the  focal  nets  of  this  congruence,  say  N,  projects 
into  the  net  N'.  Now  the  solutions  x'^'^  of  (1)  are  coordinates  both  of  N 
and  of  N  and,  with  6,  play  the  same  role  in  both  spaces  in  forming  the 
coordinates  Xo,  /*^  of  No,  i  and  Xo,  i.  But  in  order  to  form  the  last  co- 
ordinates of  A^o,  1,  namely  ^o.  i,  there  must  be  an  (n  +  l)st  coordinate  of 
A",  a  solution  of  (1)  which  may  be  called  6'. 

Again  the  third  Levy  sequence  depends  on  a  solution  ^o,  2  of  (47)  for 
its  formation.  The  argument  of  the  last  paragraph  then  demands  as  a 
necessary  and  sufficient  condition  for  the  existence  of  this  solution  a  second 
solution  ^0,  1  of  (45)  not  linearly  dependent  on  those  already  obtained. 
In  the  same  manner,  ^0,  1  calls  for  a  third  solution,  say  6",  of  (1)  not 
linearly  dependent  on  the  solutions  already  used,  such  that 


1 


'0, 1 


The  final  effect  of  this  argument  is  to  base  the  A'th  or  general  sequence 
on  k  solutions,  6,  6',  •  •  •,  d'-^~'^^  of  (1)  such  that  there  is  no  linear  relation 
between  them  and  the  .t's. 

For  further  developments,  we  must  prove,  as  a  lemma,  a  property  of 
determinants.     Consider  the  general  determinant  of  the  ?ith  order 


D  =   \ai,. 


I,  m  =  1,  2, 


Subtract  from  each  element  of  the  iih  row  the  product  of  the  corre- 
sponding element  of  the  {i  —  l)st  row  bj'  a,,  i/ai_i,  1,  (i  =  n,  ?i  —  1,  •  •  •,  2,) 
and  develop  the  result  by  the  elements  of  the  first  column.     We  have 


D  =  ai,i 


1 


«i,i 


^1, 1    ^1, 2 

tto,  1       ^2,  2 


J. 

fll,  1 


^2,  1       C(2,3 


_J^_|«1,1       «1,  n 
(2l,  1     Cto,  1       (^2,  n 


1  dn-l,  1       On— 1,2 

Ctn—1.  1  !  Cln,  1  Cln,  2 


1 


Cln-1,  1 


(ln—1,  1       ^n-1,  n 
Ctn,  1  (^n,       n 


=  ai.iA 


where  A  is  of  order  n  —  1,  so  that 


(53) 


A  =  —  D. 

rtii 


The  coordinates  of  A^o,  2  are 


^0, 2  — 


'0, 1 


^0.  1     ^*o,  1 
"1,  2     ^1, 2 


Using  r  =  0,  1  in  (46)  and  the  analogous  expressions  for  ^0. 1  and  ^1, 2  the 
coordinates  Xo,  2  become  determinants  of  the  form  of  A.     On  applying  the 


PERIODIC    CONJUGATE    NETS. 

property  expressed  in  (53)  to  them,  we  find 


252 


1 


•'^0,  2 


'0.1 


6'      X 


1 


h     6.'     x.\       ^^''' 


the  latter  expression  being  an  abbreviated  form  in  which  only  the  elements 
of  the  main  diagonal  are  shown. 

Consider  Xo,  k,  the  coordinates  of  the  net  N'o,  k-     By  definition 


^0,  it    — 


1 


Then  by  (53) 


'0,   A--1 
1 


'0,   A--1       •'^0,   A:-l 
'l,   A;  ^'l,   k 


^0,   k    —    a  a  I  "o,  fc-2  "l,  A--1  ^2,  fc  I  , 

"O,   A--2"0,   fc-l 


and  by  its  repeated  use 

^0,  k    = 


'O,  iCO,  2     •  •  •     t^O,   fc-1 


h'd.:'  ■  ■  ■  e^tl'xk 


As  this  method  of  exhibiting  the  coordinates  of  the  nets  of  the  Levy 
sequences  is  a  purely  algebraic  matter,  we  have  at  once, 


(54) 


Xr,  r+k 


1 


h"r,  T+1    • 


'r,  r+k—1 


'rUr+l 


Ur  +  k-lXr+k 


where  r  may  be  any  positive  or  negative  integer,  or  zero.  We  shall  call 
the  determinant  in  the  above  equation  Xr,  r+k',  a  determinant  like  it  but 
for  the  last  column,  in  which  the  Laplace  transforms  of  x  are  replaced  by 
those  of  d^^'\  a  {k  +  l)st  solution  of  (1),  we  shall  call  Or,  r+k-     Then 


(55) 


1 


Cr,  r+k 

From  (54)  and  (55),  we  have 

(56)  drdr,  r+1    ' 

and 

(57)  drdr,  r+l 


e 


r'Jr,  r+l 


'r,  r+k-l 


r,  r+k- 


r+k—l^r,  r+k    —    Xr,  r+k, 
%,  ?■+ A;    —    "r,  r+k- 


These  equations  are  valid  for  any  integral  value  of  r,  and  for  any  positive 
integral  value  of  k.  If  we  replace  k  in  (57)  by  /j  —  1  and  use  the  result 
in  (56)  and  (57),  we  get 

(58)  Qr,  r+k-\Xr,  r+k    =    Xr,r+k 

and 


e. 


,  r+k—l^r,  T+k 


=  G 


r,  r+k- 


Since  the  Xr,r+k  are  proportional  to  the  Xr,r+k,  the  former  may  serve 
equally  well  as  homogeneous  coordinates  of  the  nets  Nr,  r+k- 


253  EDWARD   S.    HAMMOND. 

In  the  preceding  paragraph  we  have  used  solutions  6  Unearly  inde- 
pendent of  the  coordinates  x.  The  following  theorem  states  the  situation 
under  the  opposite  condition. 

//  a  solution  6  of  the  equation  (1)  used  in  the  formation  of  any  Levy 
sequence  be  linearly  dependent  on  the  coordinates  x,  all  the  nets  of  this  Levy 
sequence  lie  in  (i^r-  2)  space;  ifi  such  solutions  he  used,  in  (j^  —  ^  —  1)  space. 

For,  supposed  =  Y.]='\g^^^x'-'^  where  the  g'-'^  are  constants  not  all  zero; 
then  dk  =  Hlz.'lg'-'^Xk'''^  for  every  k.  Now  using  these  values  of  the  Laplace 
transforms  of  6,  we  have 

i:^(^)Z<e.+.  =   |M:+i  •••  0!r^}.r^g^^^xr+k\  =  0, 

1=1 

since  the  first  and  last  columns  are  identical,  that  is,  the  coordinates  of 
all  nets  of  the  sequence  N'r,r+k  satisfy  the  equation  of  the  hyperplane 
Y,\="g''^^z'-'^  =  0,  where  the  z'^'^  are  current  coordinates.  We  observe  that 
if  all  the  g'-^^  but  one,  say  g'^^'\  are  zero,  the  nets  lie  in  the  coordinate  hyper- 
plane x^^'>  =  0.  If  6'  =  T,'i^ih'-'^x'''\  then  by  the  above  argument,  the  nets 
of  the  sequence  lie  also  in  the  hyperplane  ^\Zih'^'^z^''>  =  0.  Thus  they  lie 
in  the  intersection  of  two  hyperplanes,  or  in  space  of  order  |^^  3.  This 
proof  may  be  extended  to  the  case  stated  in  the  theorem. 

Consider  the  coordinates  of  the  net  N-s.  r',  r,  s  >  0,  in  the  form 

■^  —  s,  r    =     I  d—sd_s+l    '  '  ■    6'r-l  Xt  I  , 

where  we  have  written  only  the  elements  of  the  main  diagonal.  For  each 
of  the  Laplace  transforms  occurring  in  these  determinants  substitute  their 
values  as  linear  functions  of  the  derivatives  of  the  ^'s  and  the  o-'s  from  (21) 
and  (24).  By  suitable  operations  on  the  rows,  the  determinants  may  then 
be  reduced  to  the  form 


X.—s.r    — 


du'  du'-^  •  •  '  d  '    •  •  •      ^^^_i      ^^^ 


In  this  form  the  identity  of  the  X_s,  r  with  the  coordinates  of  the  derived 
nets  of  higher  order  k  as  defined  b\"  Tzitzeica  is  obvious.  There  are  A:  +  1 
derived  nets  of  order  k;  for  example,  the  Levy  transforms  A^_i,  o  and  A^o,  i 
are  the  derived  nets  of  the  first  order;  the  nets  N-o,  o,  N-i,  i,  and  A^o,  2 
are  the  derived  nets  of  order  two,  and  so  for  higher  orders.  We  note 
especially  that  A^_i,  1  is  the  derived  net  of  A^  depending  on  6  and  6'  in  the 
restricted  use  of  that  term;  A'',  in  turn,  is  the  derivant  net  of  A"".  Extending 
this  term,  we  say  that  A^  is  a  derivant  net  of  all  nets  Ns,  r]  r,  s  >  0. 

5.  Periodic  Levy  sequences.  If  a  sequence  of  Laplace  is  of  period  p, 
and  in  (p  —  1)  space,  we  shall  now  develop  certain  conditions  under  which 
its  Levy  sequences  have  this  same  period.     If  the  first  Levy  sequence  is 


PERIODIC    CONJUGATE    NETS.  254 

to  be  periodic,  we  must  have 


'P.  p+ 


1    —    X.To,  1, 


where  X  is  an  undetermined  factor  of  proportionality.  By  the  use  of  (37) 
and  the  value  of  Xp+i  found  by  differentiating  (37)  with  respect  to  v,  we 
obtain 


m  dp 

X 

X 

9 

X 

Op  dp^i 

Xi 

~  6 

Bx 

Xi 

that  is, 

f7ndp+i      \di\ 
xi{m  -\)  -  xy  —^ d' )  ^ 

Now  there  are  at  least  three  coordinates  x,  and  accordingly  the  coefficients 
of  x  and  .Ti  must  be  zero.  We  have  w  =  X,  and  dp+ijdp  =  6i/d,  which, 
because  of  the  definition  of  dp+i  and  di,  and  equation  (36),  becomes 

(59)  |l°g7  =  0- 

Let  us  now  differentiate  Xp,  p+i  =  inxo,  i  with  respect  to  u.  By  applying 
formulas  already  derived  for  such  derivatives,  we  get 

Xp-i,  p  =  m.r_i,  0- 

In  this  case,  using  (37)  and  (38)  and  expanding,  we  have  Hp^idp-i/dp 
=  d-i/d.     Then  (2)  and  (23)  give 

(60)  |,'o4'  =  0- 
In  consequence  of  (59)  and  (60),  we  see  that 

dp  =  Cid, 

where  Ci  is  a  constant  whose  value  is  to  be  studied  further.  To  show 
that  under  this  condition  the  nets  Np-i,  p-i+i  and  N-i,-i+i  are  identical, 
consider  the  coordinates 

1     I  ^      •         r      •      I 

^p— 1,  p—i+l    =   a         \  a  „ 

"p—i\Op—i+l       Xp_i+i  1 

Since  (41)  and  (42)  are  true  for  6  if  7n  be  replaced  by  Ci,  we  have 

Xp—i^  p—i-\-i  =  77iHHi  •  •  '  H p—iX—i,  —i+i. 

For  the  second  Levy  sequence  also  to  be  periodic,  it  is  necessary  and 
sufficient  that  do,  i  the  solution  of  (45)  by  which  the  second  Levy  sequence 
is  formed  from  the  first,  shall  be  such  that 

dp,   p+l    =    Codo,  1. 


200 


EDWARD    S.    HAMMOND. 


By  referring  to  equation  (52),  we  see  that  this  will  be  the  case  if  Qp  =  C\B' . 
In  general,  we  conclude  that  if  the  {k  —  l)st  sequence  is  periodic,  the 
necessary  and  sufficient  condition  for  the  Aih  sequence  to  be  periodic,  is 
that 

^p,  p+^-l    =    C  kdo,    k—l 

and  that  this  condition  will  be  fulfilled  if  6,  6',  •  •  •  ^^^""^^  are  such  that 
their  2;th  Laplace  transforms  are  constant  multiples  of  them.  Evidently 
under  these  last  conditions  not  only  is  the  A-th  Levy  sequence  periodic, 
but  also  all  the  sequences  of  order  less  than  k. 

The  disposition  of  the  constant  multipliers  in  various  ways  leads  to 
some  interesting  results.  By  the  last  theorem  of  section  2,  there  are  p 
solutions  6  for  every  value  of  the  constant  occurring  in  (37).  First,  let 
us  suppose  that  Ci  is  equal  to  m.  Then  6  must  be  a  linear  combination  of 
the  a-'s  and  therefore  the  theorem  of  section  4  applies  and  the  nets  of  this 
periodic  Levy  sequence  lie  space  of  order  p  —  2.  This  result  was  noted  by 
Tzitzeica.  For  sequences  of  higher  orders,  it  may  be  generalized  into  the 
following  theorem: 

//  a  sequence  of  Laplace  of  period  p  lie  in  (p  —  1)  space,  arid  has  co- 
ordinates such  that  Xp  =  ?fix,  and  if  a  Levy  sequence  of  order  k,  periodic  or 
not,  based  on  this  sequence  of  Laplace,  he  formed  by  the  use  of  k  solutions  6 
of  the  original  point  equation,  of  which  one  is  such  that  dp  =  mO,  the  nets  of 
this  Levy  sequence  lie  in  space  of  order  p  —  2;  if  i  such  solutions  be  used, 
the  Levy  sequence  is  in  space  of  order  p  —  i  —  1. 

To  prove  this  theorem,  we  need  first  to  recall  that  there  are  but  p 
solutions  of  the  system  of  partial  differential  equations  satisfied  by  the 
coordinates  of  a  periodic  sequence  of  Laplace;  therefore,  if  dp  =  7nd,  d  is 
a  linear  function  of  the  coordinates  x.  The  proof  is  then  completed  by  the 
application  of  the  last  theorem  of  section  4. 

Consider  now  the  Levy  sequences  which  can  be  formed  on  a  periodic 
sequence  of  Laplace  by  the  use  of  the  set  of  p  solutions  6,  6',  •  •  • ,  ^(p-i> 
such  that  dp'^'^  —  7n'6'''\  m'  =t=  in.  There  will  be  p  periodic  first  Levy 
sequences,  p{p  —  l)/2  periodic  second  Levy  sequences,  in  general,  as 
many  of  the  A'th  order  as  the  number  of  combinations  of  p  things  taken  A- 
at  a  time  and  finally,  one  periodic  sequence  of  the  pth  order.  It  will  now 
be  shown  that  this  pih.  sequence  coincides  with  the  original  sequence  of 
Laplace.  For,  consider  the  coordinates  of  the  net  Nq,  p,  namely,  A'o,  p. 
We  have 


X 


0,    p 


)(;'-!) 


'  p       Up         •  •  •       [/  ^^  Xp 


6^"-'^        X 


m  u    m 


m'e^P-^^       771X 


PERIODIC    CONJUGATE    NETS.  256 

Subtracting  m'  times  the  first  row  from  the  last,  then 

-^0,  p  =  ('^^  ~  ^n  )9o,  p-iXj 

so  that  the  coordinates  Xo,  p  are  proportional  to  the  x-'s.  In  general  the 
coordinates  Xr,r+p  of  the  net  A^r,  r+p  are  determinants  such  that  in  each 
the  elements  of  its  last  row  may  be  made  all  zero  but  the  last,  which  will 
be  a  constant  multiple  of  Xr.  Therefore  Xr,  r+p  is  proportional  to  xv, 
and  the  nets  Nr,  r+p  coincide  with  the  original  Laplace  sequence. 

6.  Nets  in  relation  T  and  their  Laplace  transforms.  In  the  introduction 
a  geometric  definition  of  the  relation  T  was  given;  Eisenhart  has  shown 
that,  if  A^  be  a  net  in  relation  T  with  N,  their  analytic  relation  is  expressed 
in  the  statement  that  the  homogeneous  coordinates  x  of  A^  may  be  ob- 
tained from  quadratures  of  the  form 

dx  d  fx\  dx  d  /x\ 

where  ^  is  a  solution  of  (1)  different  from  any  x\  The  factors  r  and  a  are 
not  entirely  arbitrary  for  the  conditions  of  integrability  of  (61)  show  that 
they  must  be  solutions  of  the  equations 

dr        ,  da  da  d  b 

(62)  -7-  =  (o-  —  r)  —  log  ^ ,  ^~  =  (t-  —  0")  ^-  log  -^ , 

or  their  equivalents, 

d   .       ar       ad,       a  d    ,       ha       r   d    ,       h 

—  log  -^  =  -  --  log  - ,         ^^  log  —  =  -  ^-  log  - . 

dv     ^   6        T  dv      ^  6  du      *^  B        a  du      '^  6 

In  connection  with  the  derivation  of  the  integrability  conditions  of  (61) 
it  is  readily  shown  that  the  net  A^  has  the  point  equation 

d^x  d   ,       ar  dx   ,     d    .       ba  dx 

=  ^  log  ^  x;:  +  ^  log 


dudv       dv  ^   d   du   '    du     ^  6   dv' 

For  an  equation  of  this  special  type  in  which  the  term  involving  x  is 
missing  we  shall  use  a  symbol  similar  to  (13)  except  that  the  last  of  the 
quantities  within  the  brackets  is  omitted_to  indicate  that  the  term  in  x 
is  lacking.     Thus,  the  point  equation  of  A^  will  be  denoted  by 


(63)  [-^'T'TJ- 


Using  the  fact  that  ^  is  a  solution  of  (1),  the  invariants  of  A^  have  the  va'ues 

(64)  H  =  H-'^^,        K^K-'^f^. 

^     '  dudv  dudv 


257  EDWARD    S.    HAMMOND. 

From  these  developments,  it  appears  that  the  determination  of  a  net 
in  relation  T  with  N  depends  on  a  solution  of  (1),  a  pair  of  functions  r 
and  0- which  satisfy  (62),  and  the  quadratures  (61).  Eisenhart  has  shown 
that  the  problem  may  be  given  another  aspect  by  the  introduction  of  a 
function  0,  defined  by  the  equation 

(65)  T  -  <j  =  (f>d. 

By  differentiation  and  the  use  of  (62)  and  (1)  it  may  be  shown  that  0  is  a 
solution  of  the  equation  denoted  by 

But  this  is  the  adjoint  of  (1).  Accordingly,  the  problem  is  reduced  to  the 
finding  of  a  solution  of  (1)  and  a  solution  of  its  adjoint  equation,  and  two 
sets  of  quadratures,  namely 

dr  d  log  b(h  dr  ^  d   .       a 

—  =  00  — ^^     - ,         —  =  -  00  —  log  - , 
du  •  du       '  dv  ^    dv     ^  6' 

(67) 

da  ^^   d   .      b  da  5  log  a0 

-TT-  =  00  ^r- log  -  ,  -T-   =    —  00 7. , 

du  ^    du     ^  6'  dv  dv       ' 


which  follow  from  (65)  and  (62),  and  (61). 

A  discussion  of  the  effect  on  the  net  N  of  the  arbitrary  constants  arising 
from  these  quadratures  is  in  order  at  this  time.  If  x  be  the  coordinates 
of  the  net  A'  when  the  additive  constant  ^To  r  and  a  is  set  equal  to  zero,  then 
for  any  other  value  of  c,  the  coordinates  of  the  T  transform  become  X-  +  ex;  0. 
This  point  is  on  the  line  joining  corresponding  points  of  A"  and  N.  Conse- 
quently, we  may  say  that  the  variation  of  this  constant  leaves  the  conju- 
gate congruence  of  the  transformation  unchanged  but  moves  the  points 
of  the  net  along  the  lines  of  this  congruence. 

Again  if  x'^'^  and  x'-'^  +  c,  are  the  coordinates  of  nets  obtained  by 
different  values  of  the  constant  of  integration  in  (61),  the  line  of  inter- 
section of  the  tangent  planes  to  the  nets  is  the  same  for  all  values  of  d. 
This  is  a  result  of  equation  (61)  since  the  coordinates  of  the  Lev}^  trans- 
forms of  N  by  means  of  0  may  be  taken  as  d{x/6)idu  and  d{Xjd)jdv. 
The  totality  of  such  lines  of  intersection,  or  the  joins  of  corresponding 
points  of  the  Levy  transforms  form  a  congruence  which  has  been  termed 
by  Guichard*  the  harmonic  congruence  of  the  transformation.  We  may 
say  then,  that  the  variation  of  the  constant  arising  from  (61)  leaves  the 
harmonic  congruence  of  the  transformation  unchanged.  Conversel}',  all 
nets  harmonic  to  this  congruence  are  so  determined,  since  it  has  been 

*  Guichard,  Annales  de  I'Ecole  Normale  Sup.,  3^  Serie,  t.  14  (1897),  p.  4S3. 


PERIODIC    CONJUGATE    NETS.  258 

shown  by  Eisenhart*  that  two  nets  harmonic  to  a  congruence  are  in 
relation  T. 

Now  if  Xi  and  di  be  the  first  Laplace  transforms  of  x  and  6,  the  co- 
ordinates of  A^i,  a  T  transform  of  A^i  will  be  given  by  quadratures  similar 
to  (61),  namely, 

The  integrability  conditions  of  this  quadrature  give  equations  for  n  and 
cTi  analogous  to  (62), 

"^  =  (o"!  —  •^i)  -T"  iog  ^fl">        ii     =  (^1  ~  (^i)  T~  log  ;r 

dv  dv      ^   ^1  ^2*         ^  ^  du      ^  01 

and  we  also  find  that  the  point  equation  of  A'^i  is  denoted  by 

(69)  L^-^'-^'  7rJ- 

In  order  to  determine  the  relation  between  ri,  o-i  and  r,  o-,  we  proceed 
as  follows.  If  A^i  is  the  first  Laplace  transform  of  A^,  the  invariants  Hi 
and  H  of  these  nets  should  be  related  as  are  the  invariants  H^  and  H  in 
(12).     Forming  the  corresponding  relation,  we  have 

-  d-  gtH        - 

Hi  =  —  ^—^  log  -T h  H. 

dudv      ^    OCT 

On  reducing  this  equation  by  the  use  of  (69),  (64),  and  (12),  we  find  that 

Similar  reckoning  performed  with  Ki  and  K  shows  that 

d'^  log  di       d'  log  r 


(71) 


dudv  dudv 


Now  the  equations  denoted  by  (63)  and  (69)  are  of  the  form  which 
must  be  satisfied  by  the  non-homogeneous  coordinates  of  a  net,  and  this 
suggests  that  the  same  relation  may  hold  between  the  coordinates  of  Ni 
and  A^  that  holds  in  the  non-homogeneous  case.  This  is  shown  to  be 
true,  for  if  we  substitute 

-    _  -  _         1         dx 

Xi  X  ^  1      > 

d  ,       ar  dv 


■  A  result  as  yet  unpublished. 


250  EDWARD    S.    HAMMOND. 

and 


T 


'H 


— particular  solutions  of  (70)  and  (71) — and  the  values  of  .ri  and  ^i  given 
by  (2)  and  (44)  in  equations  (OS),  they  are  identicallj^  true.  We  have 
proved  then  that  the  T  transforms  of  a  net  A''  and  its  first  Laplace  trans- 
form whose  coiirdinates  are  obtained  from  the  quadratures  (01)  and  (OS), 
where  ^i  is  the  first  Laplace  transform  of  6,  and  where  n  and  ai  have  the 
above  values,  are  Laplace  transforms  of  one  another. 

We  find  that  the  difTereuce  n  —  ci,  when  reduced  by  the  use  of  (04) 
and  (07),  is  equal  to  -  0_i  dilH,  where 

d4>      d  log  b 

0-1  ~  ~^ — —  '\       4>f 
^  da  dii        ' 

following  (2)  and  (00).     Now-  </)_i  satisfies  the  equation  denoted  by 

r     1  //  1 1 

and  0_i/7/,  because  of  (17),  satisfies 

V4>.,    J_    1      1    ] 
IH  '  aH'  b'  a'Hi' 

But  this  equation  is  the  adjoint  of  (10);  so  that  we  have  the  net  .Yi 
based  on  a  solution  of  the  adjoint  of  the  point  equation  of  .Vi  which  is 
proportional  to  the  minus  first  Ltiplace  transform  of  0. 

In  general,  we  have  that,  if  A'r  is  a  T  transform  of  .Yr,  the  rth  Laplace 
transform  of  A",  whose  coordinates  are  given  by  the  quadratures 

then  the  nets  vYr  (/*  =  0,  1,  2,  •  •  •),  form  a  sequence  of  Laplace.  In  this 
general  case,  w'e  find,  as  in  the  particular  cases  we  have  considered,  that 
</)'_r,  defined  by 

Tr    —    (Tr 

is  a  solution  of  the  adjoint  of  the  point  equation  of  Nr,  which  is  denoted 
bv 

:  r   ,  1 1  b^'"^ ] 

(/4)  1^  0_r;  ^jjfj^  V. .  Hr-i '  b '  a^+'H-Hi^'  •  •  •  Hr-i  J  * 

But  using  equation  (25)  we  may  denote  the  equation  of  Laplace  satisfied 


PERIODIC   CONJUGATE   NETS.  260 

l)y  0_r,  dofi  110(1  as 


d(j)-r+\   ,     d  b 

But  tlio  (iiKiiilily  7///1  •  •  •  y7,_i0'_,-,  bocauso  of  (71)  and  (17)  also  satislics 
this  cqiuilioii,  mid  therefore  we  have 

Tr    —    (Tr  ,,  ^-r 


dr  HHi    '  '  '   Hr—1 

Therefore  the  quadra  lures  to  determine  t,  and  o-,,  corresponding;  to  (67) 
are 

dTr  0,(f^-r  d  h(l)-r 

du   "  lllh  ■  ■  ■  Hr-i'd^^'^IIHi  ■  ■  ■  //,_:' 


(70) 


dTr  _    _  OrCfy,-  d_.         (ill    ■  ■  ■    Ilr-l 

'dV     ~  Illh       ■'   Hr-ldV^'''^  Or 

d(Tr  dr<t>—r  d    .         0 

no'  — 


du        IIUx  ■  •  •  Hr-1  du 

r»     =  —  jjTT  jj      3-loga0_,.. 

dv  HH X  •  •  •  Iir-1  dv     ^ 

7.  Periodic  sequences  of  T  transforms.  As  a  preliminary  step  in  the 
question  of  the  i)eriodieity  of  the  T  tninsforms,  we  investig;ate  the  adjoint 
equation  of  (1)  when  (34)  and  (36)  are  satisfied.  These  equations  are 
necessary  and  sufhcient  conditions  for  a  periodic  sequence  of  Laplace 
whose  coordinates  satisfy  (1)  and  (37).  Now  if  the  invariants  of  (66) 
are  foimed,  it  is  found  that  they  are  the  same  as  H  and  K  but  are  inter- 
changed. Similarly  the  invariants  of  (72)  are  those  of  (16),  that  is, 
Hi  and  7vi,  but  intei-clianged;  and  in  general,  the  invariants  of  (75)  are 
II r  and  Kr  interclianged.  We  also  notice  that  in  working  with  (72),  a 
and  h  are  replaced  by  1/a  and  1/6,  respectively.  Then  the  conditions  on 
the  coefficients  of  (()6)  which  assure  solutions  such  that 

are  equivalent  to  (34)  and  (36),  and  we  may  state  the  following  theorem: 

If  an  cqualion  of  Laplace  has  periodic  solutions,  so  has  its  adjoint. 

Suppose  that  the  fundamental  sequence  is  periodic  of  period  p,  and 

that  Xj,  =  7nx,  the  conditions  (34)  and  (36)  being  satisfied.     Let  0,  the 

solution  of  the  adjoint  equation  of  (1)  which  determines  the  quadratures 


261  EDWARD    S.    HAMMOND. 

(67)  and  (61),  be  such  that  0  =  n0_p;  also  let  the  solution  ^  of  (1)  involved 
in  these  quadratures  be  such  that  dp  =  m'd.  Then  if  we  set  r  =  p  the 
quadratures  (73)  and  (76)  which  determine  a  pth  Laplace  transform  of  A'^ 
become 


dtp      m'        b  .      ^  Bt-p  w'  ^     5  ,      a 

— -P  =  —  ^0  —  log  60,  —^= ^0      log-, 

d(Tp       m'         d  ,      h  dcTp  m'        d  . 

"a  ~  =  —  ^0  -^  log  T ,  ^-"  = ^0  —  log  a4>, 

du        n        du     ^  6  dv  n     ^  dv     ^    ^' 


(79) 

and 

,     .  dxp  _m      d  / x\  dxp  _m     d  fx\ 

^^  'du  ~  n'"du\d)'         ^  ~n'^&u\d)' 

which  differ  from  (67)  and  (61)  only  by  constant  factors.  Then  the 
coordinates  Xp  can  differ  from  x  only  by  a  constant  factor  arising  from  the 
factors  appearing  in  (79)  and  (80),  or  by  an  additive  constant  from  the 
final  quadratures.  But  these  additive  constants  are  entirely  arbitrary, 
and  since  we  are  dealing  with  homogeneous  coordinates  the  factor  is 
immaterial. 

We  have,  now,  the  following  theorem:  Let  (1)  he  the  point  equation  of 
the  fundamental  net  N  of  a  sequence  of  Laplace  of  period  p,  whose  coordinates 
X  are  such  that  Xp  =  mx]  if  6  he  a  solution  of  (1)  such  that  dp  =  7n'd,  and  if 
({)he  a  solution  of  the  adjoint  equation  of  (1)  such  that  0  =  n(t)^p,  then  each  T 
transform  of  N  determined  hy  quadratures  from  6  and  4)  is  the  fundamental 
net  of  a  sequence  of  Laplace  of  period  p;  moreover,  each  of  the  nets  of  these 
sequences  is  a  T  transform  of  the  corresponding  net  of  the  original  sequence. 

The  author  wishes  to  express  his  gratitude  to  Professor  Eisenhart  for 
the  suggestions  which  led  to  this  paper  and  for  continued  helpful  advice 
and  criticism  during  its  preparation. 


Gaylord  Bros. 

Makers 

Syracuse,  N.  Y, 

PAT,  JAN,  21,  1903 


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